The quantum system with an interaction between a closed Y-type four-level atom and multi-mode light fields is adopted to possess left handedness by means of the technique of quantum coherence. In the representation of interaction, the density matrix method is utilized in view of the rotating-wave approximation and the di-pole approximation.The conclusion of the numerical simulation shows that under the appropriate parameter conditions, the medium can achieve negative permittivity and negative permeability simultaneously (i.e. the left handedness effect happens and the left-handed material is realized.) In the condition of the weak probe field, when the Rabi frequency of probe light increases, the frequency range of the medium to achieve its left-handed effect decreases and the refractive index and absorption coefficient will also changed correspondingly. The medium to affect light absorption and gain in the conditions of the establishment of left -handed effect are also discussed. The paper can be concluded that, in the conditions of weak probe field, the electro-magnetic induction can be used to achieve the left-handed effect of the medium, which extends the implementation of left-handed materials in the experiment, and extends the application field of electro-magnetic induction.